Proving the impossibility of a proof

Question

Given that according to Gödel's theorems there are propositions in any language equivalent to first order logic that cannot be proven right or wrong, is it possible to prove that some of such propositions are not provable? I am thinking of something like a proof of impossibility applied to propositions.

Answer 1

Gödel showed how to construct a statement in any sufficiently powerful formal system that states its own unprovability within the language of the formal system itself (he did this by showing that propositions in the language of the formal system can be translated into numbers and that "provability" can be translated into artithmetical properties of those numbers). But his construction method creates a specific proposition - it cannot be used to determine the provability of a general proposition.

This is analagous to how Liouville showed that transcendental numbers exist by constructing a family of numbers with properties which allowed him to prove they were not algebraic. It is much more difficult to prove that a particular number such as $e$ or $\pi$ is transcendental.

Answer 2

Impossibility of a proof in generality means: the statement is false.

In mathematics all true statements are provable.

Considering mathematics as an abstract model of certain aspects of reality, a statement being true means: it's consistent with all other statements of the model.

If a statement is consistent, there are ways to show it, to construct a proof.

As for Gödel, something not provable "within the language of the formal system itself", that will happen, as no sentence may prove themselves.

Translating Gödels argument into building a house, into construction, may read:

When building a house, the correctness of its construction might may not be stated completely from within.

Correct. But, why not go outside and check it from some distance?